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Consider a 2-D square array of qubits of extent L × L. We provide a proof that the minimum weight perfect matching problem associated with running a particular class of topological quantum error correction codes on this array can be exactly solved with a 2-D square array of classical computing devices, each of which is nominally associated with a fixed number N of qubits, in constant average time per round of error detection independent of L provided physical error rates are below fixed nonzero values, and other physically reasonable assumptions. This proof is applicable to the fully fault-tolerant case only, not the case of perfect stabilizer measurements.

Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time